Killing (super)algebras for generalised spin manifolds (2511.07246v1)

Abstract: We define the notion of a Killing (super)algebra for a connection on a spinor bundle associated to a generalised spin structure on a pseudo-Riemannian manifold of any signature. We are led naturally to include in the even subspace not only Killing vectors but also certain infinitesimal gauge transformations, and we show that the definition of the (super)algebra requires, in addition to the spinor connection and a Dirac current, a map to pair spinor fields into infinitesimal gauge transformations. We show that these (super)algebras are filtered subdeformations of (an analogue of) the Poincaré superalgebra extended by the (R)-symmetry algebra. By employing Spencer cohomology, we study such deformations from a purely algebraic point of view and, at least in the case of Lorentzian signature and high supersymmetry, identify the subclass of deformations to which the Killing superalgebras belong. Finally, we show that, with some caveats, one can reconstruct a supersymmetric background geometry from such a deformation as a homogeneous space on which the deformation is realised as a subalgebra of the Killing superalgebra.

• The paper introduces a framework for constructing Killing superalgebras by integrating twisted spin structures with R‑symmetry.
• It employs filtered subdeformations and Spencer cohomology to systematically classify admissible algebraic deformations.
• The study provides a unified approach applicable to the classification of supersymmetric backgrounds in supergravity and computational geometry.

Killing (Super)algebras on Generalised Spin Manifolds with R-Symmetry

Introduction

This paper establishes a comprehensive framework for Killing (super)algebras associated to connections on spinor bundles over pseudo-Riemannian manifolds possessing generalised spin (spin-G) structures, focusing primarily on the inclusion of $R$-symmetry. The formalism generalises previous investigations into Killing superalgebras, which are central to the algebraic underpinning of supersymmetry in supergravity and related geometric structures, by accommodating twisted spin structures and their attendant gauge-theoretic features. The even part of these superalgebras incorporates both Killing vectors and infinitesimal $R$-symmetry gauge transformations, while the odd part comprises appropriately defined Killing spinors. The construction utilises filtered subdeformations of $R$-symmetry-extended flat model superalgebras (generalised Poincaré superalgebras) and leverages Spencer cohomology for a classification of their algebraic deformations.

Generalised Spin Structures and R-Symmetry

A generalised spin structure, or spin-$G$ structure, extends the classical notion of a spin structure by allowing a "twist" via an internal symmetry group $G$ with a central $\mathbb{Z}_2$ subgroup. The structure group of the oriented frame bundle is lifted from $\mathrm{SO}(s,t)$ to $$\mathrm{Spin}^G(s,t) = \mathrm{Spin}(s,t) \times_{\mathbb{Z}_2} G$$. This generalisation admits physically relevant fermionic couplings in cases where pure spin structures are topologically obstructed. Examples include spin-$c$ ($G = U(1)$), spin-$h$ ($G = SU(2)$), and more recent geometric interest in spinorially twisted (spin-$k$) structures ($G = \mathrm{Spin}(k)$).

The $R$-symmetry group, defined as the automorphisms of the spinor representation preserving the Dirac current $\kappa$, acts naturally by graded automorphisms on the superalgebras built from these structures, with effective action on the odd part and trivial action on the even part. The corresponding extension of the flat model superalgebra by the $R$-symmetry algebra, matching the structure of supersymmetry algebras in supergravity, is formalised at both the Lie algebra and Lie superalgebra level.

Killing (Super)algebra Construction

Symmetry Algebra

The symmetry algebra $symm(\mathcal{S},A)$ for a generalised spin structure with connection $A$ is formed by combining the Lie algebra of Killing vectors $iso(M,g)$ and infinitesimal $R$-symmetry gauge transformations $R = \Gamma(\mathrm{ad}\,Q)$, but endowed with a bracket $[\cdot,\cdot]$ twisted by curvature terms $F$ of the $R$-connection. The resulting extension encodes both geometric and gauge symmetries, manifesting as an extension rather than a direct sum.

Killing Spinors and Superalgebra Definition

A Killing spinor is a section of the twisted spinor bundle $\mathcal{S}$ annihilated by a specified connection $D$, which differs from the Levi-Civita connection by a bundle endomorphism $\beta$. The Killing (super)algebra $\mathfrak{k}_{(D,\rho)}$ is canonically built from:

• Even part: Killing vectors annihilating $\beta$, $R$-symmetries parallel under $D$, additionally required to preserve the background data ($\beta$, $\rho$).
• Odd part: $D$-parallel spinor fields (Killing spinors).

The bracket on pairs of spinors generically yields both a Killing vector (via the Dirac current) and an infinitesimal $R$-symmetry (via a background-dependent pairing $\rho$). Closure under the Lie (super)algebra relations, particularly the Jacobi identities, imposes stringent compatibility constraints on $\beta$ and $\rho$, generalising the admissibility conditions for connections in classical Killing superalgebra theory.

Filtered Subdeformations and Spencer Cohomology

A central result is that Killing (super)algebras for generalised spin manifolds with $R$-symmetry are filtered subdeformations of the $R$-extended flat model superalgebra. The structure of possible deformations is governed by the Spencer (2,2)-cohomology of the graded subalgebra defined by the geometric data. Highly supersymmetric cases ($\dim S' > \frac{1}{2}\dim S$) in Lorentzian signature exhibit key simplifications: surjectivity of the Dirac current and transitivity of the action, enabling faithful $R$-symmetry representation and facilitating deformation classification.

Filtered deformations are parametrised by cocycles $(\alpha, \beta, \gamma, \rho)$ satisfying Spencer cocycle conditions, mixed invariance under the even subalgebra, and integrability via higher order Jacobi identities. For geometrically realisable deformations, only certain classes of cocycles are admissible, leading to deformations where nontrivial $R$-symmetry field strengths vanish, matching the restrictive structure of supergravity symmetry algebras.

Homogeneous Backgrounds and Reconstruction

When the graded subalgebra is highly supersymmetric and geometrically realisable, a reconstruction theorem demonstrates that each filtered subdeformation yields a corresponding homogeneous spin-$R$ manifold equipped with an admissible pair $(D, \rho)$ and an embedding of the superalgebra into the Killing superalgebra of the geometry. This is achieved via the theory of homogeneous spin-$R$ structures, extension of Wang's theorem for invariant connections, and the formal machinery of super Harish-Chandra pairs. The background's curvatures and field strengths are explicitly tied to the deformation cocycles, while the geometric data is encoded as homogeneous bundles and connections.

Implications and Prospects for AI and Mathematical Physics

The established formalism provides a unified approach to constructing and classifying symmetry superalgebras for geometric and physical backgrounds with internal symmetries, especially those relevant to gauged supergravity. The explicit Spencer cohomology perspective enables systematic identification of admissible deformations, potentially guiding the classification of allowed supersymmetric backgrounds, their moduli, and the algebraic underpinning of their symmetry properties.

In computational geometry and AI-driven symbolic systems, these results inform automated reasoning about algebraic and geometric structures on manifolds. Algorithms for checking admissibility, cohomology computations, and deformation integration can be directly formulated, facilitating machine-assisted derivation and verification of superalgebraic models in both mathematical and physical contexts.

The generality and flexibility of the construction suggest promising directions for future development: relaxing certain constraints may allow for richer symmetry algebras, especially those with dynamical $R$-symmetry fields, and the interplay with higher gauge theory could provide further generalisation. Applications to geometric realisation of exceptional Lie algebras, classification of supersymmetric solutions in supergravity, and analysis of quantum field theoretic invariants are immediate.

Conclusion

The paper gives a complete and technically detailed framework for Killing (super)algebras associated to generalised spin manifolds with $R$-symmetry, encompassing both geometric and algebraic aspects. By leveraging filtered subdeformations, Spencer cohomology, and homogeneous space constructions, it generalises the standard supergravity symmetry algebra formalism, enabling rigorous analysis and classification of supersymmetric backgrounds and their symmetry structures in a broad geometric setting. The theoretical apparatus constructed here is immediately applicable to a wide range of problems in mathematical physics and physical geometry, and offers a solid foundation for future computational and categorical approaches to symmetry in geometry and field theory.